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Pythagorean Theorem Distance and
Midpoints
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Distance Formula
Midpoints
Table of Contents
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Pythagorean Theorem
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Pythagorean Theorem
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Pythagorean Theorem
This is a theorem that is used for right triangles. It was first known in ancient Babylon and Egypt beginning about 1900 B.C. However, it was not widely known until Pythagoras stated it.
Pythagoras lived during the 6th century B.C. on the island of Samos in the Aegean Sea. He also lived in Egypt, Babylon, and southern Italy. He was a philosopher and a teacher.
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Legs
- Opposite the right angle- Longest of the 3 sides
- 2 sides that form the right angle
click to reveal
click to reveal
click to reveal
Labels for a right triangle
ca
b
Hypotenuseclick to reveal
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In a right triangle, the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c).
a2 + b2 = c2
Link to animation of proof
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a2 + b2 = c2
52 + b2 = 152
25 + b2 = 225
-25 -25
b2 = 200
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer5 ft
15 ft
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9 in18 in
a2 + b2 = c2
92 + b2 = 182
81 + b2 = 324
-81 -81
b2 = 243
Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
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4 in
7 in
a2 + b2 = c2
42 + 72 = c2
16 + 49 = c2
65 = c2
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root &Label Answer
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Missing Leg
Write Equation
Substitute in numbers
Square numbers
Subtract
Find the Square Root
Label Answer
Missing Hypotenuse
Write Equation
Substitute in numbers
Square numbers
Add
Find the Square Root
Label Answer
How to use the formula to find missing sides.
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1 What is the length of the third side?
4
7x
Ans
wer
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2 What is the length of the third side?
Ans
wer
41x
15
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3 What is the length of the third side?
Ans
wer
7
z4
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4 What is the length of the third side?
3
4
x
Ans
wer
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3
4
There are combinations of whole numbers that work in the Pythagorean Theorem. These sets of numbers are known as Pythagorean Triplets.
3-4-5 is the most famous of the triplets. If you recognize the sides of the triangle as being a triplet (or multiple of one), you won't need a calculator!
5
Pythagorean Triplets
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Trip
les 12 = 1 112 = 121 212 = 441
22 = 2 122 = 144 222 = 48432 = 9 132 = 169 232 = 52942 = 16 142 = 196 242 = 57652 = 25 152 = 225 252 = 62562 = 36 162 = 256 262 = 67672 = 49 172 = 289 272 = 72982 = 64 182 = 324 282 = 78492 = 81 192 = 361 292 = 400102 = 100 202 = 400 302 = 900
Can you find any other Pythagorean Triplets?
Use the list of squares to see if any other triplets work.
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5 What is the length of the third side?
6
8
Ans
wer
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6 What is the length of the third side?
513
Ans
wer
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7 What is the length of the third side?
48
50
Ans
wer
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8 The legs of a right triangle are 7.0 and 3.0, what is the length of the hypotenuse? A
nsw
er
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9 The legs of a right triangle are 2.0 and 12, what is the length of the hypotenuse?
Ans
wer
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10 The hypotenuse of a right triangle has a length of 4.0 and one of its legs has a length of 2.5. What is the length of the other leg?
Ans
wer
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11 The hypotenuse of a right triangle has a length of 9.0 and one of its legs has a length of 4.5. What is the length of the other leg?
Ans
wer
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Corollary to the Pythagorean TheoremIf a and b are measures of the shorter sides of a triangle, c is the measure of the longest side, and c2 = a2 + b2, then the triangle is a right triangle.
If c2 ≠ a2 + b2, then the triangle is not a right triangle.
b = 4 ft
c = 5 fta = 3 ft
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Corollary to the Pythagorean Theorem
In other words, you can check to see if a triangle is a right triangle by seeing if the Pythagorean Theorem is true.
Test the Pythagorean Theorem. If the final equation is true, then the triangle is right. If the final equation is false, then the triangle is not right.
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Is it a Right Triangle?
Write Equation
Plug in numbers
Square numbers
Simplify both sides
Are they equal?
8 in, 17 in, 15 in
a2 + b2 = c2
82 + 152 = 172
64 + 225 = 289
289 = 289
Yes!
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12 Is the triangle a right triangle?
Yes
No
8 ft
10 ft6 ft
Ans
wer
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13 Is the triangle a right triangle?
Yes
No
30 ft
24 ft36 ft
Ans
wer
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14 Is the triangle a right triangle?
Yes
No10 in.
8 in.12 in.
Ans
wer
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15 Is the triangle a right triangle?
Yes
No Ans
wer
5 ft13 ft
12 ft
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Ans
wer
16 Can you construct a right triangle with three lengths of wood that measure 7.5 in, 18 in and 19.5 in?
YesNo
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Steps to Pythagorean Theorem Application Problems.
1. Draw a right triangle to represent the situation.2. Solve for unknown side length.3. Round to the nearest tenth.
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17 The sizes of television and computer monitors are given in inches. However, these dimensions are actually the diagonal measure of the rectangular screens. Suppose a 14-inch computer monitor has an actual screen length of 11-inches. What is the height of the screen?
Ans
wer
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18 A tree was hit by lightning during a storm. The part of the tree still standing is 3 meters tall. The top of the tree is now resting 8 meters from the base of the tree, and is still partially attached to its trunk. Assume the ground is level. How tall was the tree originally?
Ans
wer
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19 You've just picked up a ground ball at 3rd base, and you see the other team's player running towards 1st base. How far do you have to throw the ball to get it from third base to first base, and throw the runner out? (A baseball diamond is a square)
home
1st
2nd
3rd
90 ft.
90 ft.90 ft.
90 ft.
Ans
wer
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20 You're locked out of your house and the only open window is on the second floor, 25 feet above ground. There are bushes along the edge of your house, so you'll have to place a ladder 10 feet from the house. What length of ladder do you need to reach the window?
Ans
wer
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Distance Formula
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If you have two points on a graph, such as (5,2) and (5,6), you can find the distance between them by simply counting units on the graph, since they lie in a vertical line.
The distance between these two points is 4.
The top point is 4 above the lower point.
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22 What is the distance between these two points?
Pul
lP
ull
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23 What is the distance between these two points?
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24 What is the distance between these two points?
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Most sets of points do not lie in a vertical or horizontal line. For example:
Counting the units between these two points is impossible. So mathematicians have developed a formula using the Pythagorean theorem to find the distance between two points.
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Draw the right triangle around these two points. Then use the Pythagorean theorem to find the distance in red.
c2 = a2 + b2
c2 = 32 + 42
c2 = 9 + 16c2 = 25c = 5a
bc
The distance between the two points (2,2) and (5,6) is 5 units.
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Example:
c2 = a2 + b2
c2 = 32 + 62
c2 = 9 + 36c2 = 45
The distance between the two points (-3,8) and (-9,5) is approximately 6.7 units.
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Try This:
c2 = a2 + b2
c2 = 92 + 122
c2 = 81 + 144c2 = 225c = 15
The distance between the two points (-5, 5) and (7, -4) is 15 units.
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Deriving a formula for calculating distance...
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Create a right triangle around the two points. Label the points as shown. Then substitute into the Pythagorean Formula.
(x1, y1)
length = x2 - x1
length = y2 - y1
d
c2 = a2 + b2
d2 = (x2 - x1)2 + (y2 - y1)2
d = (x2 - x1)2 + (y2 - y1)2
This is the distance formula now substitute in values.
d = (5 - 2)2 + (6 - 2)2
d = (3)2 + (4)2
d = 9 + 16
d = 25
d = 5
(x2, y2)
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Distance Formula
d = (x2 - x1)2 + (y2 - y1)2
You can find the distance d between any two points (x1, y1) and (x2, y2) using the formula below.
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How would you find the perimeter of this rectangle?
Either just count the units or find the distance between the points from the ordered pairs.
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A (0,-1)B (8,0)
C (9,4)D (3,3)
Can we just count how many units long each line segment is in this quadrilateral to find the perimeter?
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Midpoints
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(2, 2)
(2, 10)
Find the midpoint of the line segment.
What is a midpoint?How did you find the midpoint?What are the coordinates of the midpoint?
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(3, 4) (9, 4)
Find the midpoint of the line segment.
What are the coordinates of the midpoint?How is it related to the coordinates of the endpoints?
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(3, 4) (9, 4)
Find the midpoint of the line segment.
What are the coordinates of the midpoint?How is it related to the coordinates of the endpoints?
Midpoint = (6, 4)
It is in the middle of the segment.
Average of x-coordinates.Average of y-coordinates.
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The Midpoint Formula
To calculate the midpoint of a line segment with endpoints (x1,y1) and (x2,y2) use the formula:
(x1 + x2 y1 + y2
22, )
The x and y coordinates of the midpoint are the averages of the x and y coordinates of the endpoints, respectively.
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The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)
See next page for answer
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The midpoint of a segment AB is the point M on AB halfway between the endpoints A and B.
B (8,1)
A (2,5)Use the midpoint formula:
Substitute in values:
( x1 + x2 y1 + y2
22, )
2 + 8 , 5 + 12 2( )
Simplify the numerators:10 62 2
,
Write fractions in simplest form:( )
(5,3) is the midpoint of AB
M
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Find the midpoint of (1,0) and (-5,3)
Use the midpoint formula:
Substitute in values:
( x1 + x2 y1 + y2
22, )
1 + -5 , 0 + 32 2( )
Simplify the numerators:-4 32 2
,
Write fractions in simplest form:( )(-2,1.5) is the midpoint
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36 Find the center of the circle with a diameter having endpoints at (-4,3) and (0,2).
Which formula should be used to solve this problem?
A Pythagorean Formula
B Distance Formula
C Midpoint Formula
D Formula for Area of a Circle
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